Gruber P. Convex and Discrete Geometry

The Isoperimetric Inequality in a Minkowski Space
8 The Brunn–Minkowski Inequality 155
The analog of the classical isoperimetric inequality in a Minkowski space is as
follows:
Theorem 8.10. Let C ∈ Cp(E d ) where E d is endowed with a new norm and a corresponding
isoperimetrix I . Then
SI (C) d
V (C) d−1 ≥ SI (I ) d
,
V (I ) d−1
where equality holds if and only if C is homothetic to I .
If, in the three proofs of the classical isoperimetric inequality given above, the
Euclidean unit ball B d is replaced by I and instead of Steiner’s formula for the volume
of parallel bodies Minkowski’s theorem on mixed volumes is used, we obtain
this more general result.
8.4 Sand Piles, Capillary Surfaces and Wulff’s Theorem in Crystallography
As remarked Sect. 8.3, the Brunn–Minkowski theorem has several applications to
isoperimetric problems of mathematical physics.
In the following we present three pertinent results, a problem on the maximum
volume of a sand pile with given area of the base, a problem on the minimum volume
of a capillary surface and Wulff’s theorem on the form of crystals.
For general references on isoperimetric inequalities of mathematical physics, see
the books and surveys cited in the introduction to Sect. 9.4.
Sand Piles of Maximum Volume
What is the shape of a closed planar set of given area which supports a sand pile
of maximum volume? The same mathematical problem arises also in Nádai’s [761]
theory of plasticity in the following form: For what cross-section of given area has
a perfectly plastic rod maximum torsional rigidity? Compare Sect. 9.4 for the corresponding
question for elastic rods. Leavitt and Ungar [633] proved the following
result using the method of inner parallel sets. This method can also be used to prove
the geometric isoperimetric inequality, see, e.g. Fejes Tóth [329].
Theorem 8.11. Among all compact sets B in E 2 of given area it is precisely the
circular discs, up to sets of measure 0, that support sand piles of maximum volume.
Proof. Let α be the maximum angle with the horizontal which a sand pile can sustain,
the so-called glide angle of the sand. The height of a sand pile on B at a point
x ∈ B then is at most δ(x) tan α, where δ(x) = dist(x, bd B) = min{�x − y� :y ∈
bd B}. Thus we have the following estimate for the volume V of a sand pile on B:
�
(1) V ≤ tan α δ(x) dx.
B